Method of processing information to be confidentially transmitted

ABSTRACT

A method of processing information to be confidentially transmitted from a first module to a second module provides that a first scalar multiplication may be carried out in order to obtain a first result [r]P). This first scalar multiplication comprises a plurality of generation steps of ordered factors from which a plurality of first partial sums are required to be built. The method also comprises the carrying out of a second scalar multiplication in order to obtain a second result. This second multiplication provides that a plurality of second partial sums may be built. A piece of encrypted information is obtained by processing the information based on the results of said scalar multiplications. The second partial sums of the second scalar multiplication use the same ordered factors obtained by the generation step of the first scalar multiplication.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to communication protocols among modules, for example, of the electronic type. Particularly, the invention relates to a method of processing information which can be employed, preferably though not exclusively, in the protocols ensuring a safe communication among these modules.

2. Description of the Related Art

As it is known, the need for making a transmission either of information or data of any nature safe among modules either of the hardware (for example, electronic devices) or software types communicating with one another is felt more and more, within the scope of the Information Technology.

To this purpose, encryption techniques of the transmitted messages have been refined in order to scramble the latter for non-authorized people.

As it is known to those skilled in the art, the most advanced encryption techniques are based on mathematical schemes set in protocols of encrypted communication either based on the identity or of the IBE-type (Identity Based Encryption).

Moreover, the interest in the mathematical schemes being applied to elliptical curves, such as for example the schemes based on a pairing is growing within the scope of these encryption protocols. Particularly, the most employed pairing is the Tate's one.

As it is known, in accordance with some protocols based on the identity, in order to safely transmit data that we have indicated with M from a first module A to a second module B, there is provided the carrying out of two subsequent operations by the first module A.

A first operation is a scalar multiplication which is carried out on a point of an elliptical curve by a random integer generated by module A.

A second operation provides that a known pairing function, for example, the Tate's pairing is calculated.

At this point, the data M to be transmitted are processed by the first module A by means of this pairing function and are transmitted to the second module B together with the result of the scalar multiplication.

The second module B may re-ascend the data M being sent by means of a private key of the latter, starting from the information received by the first module A.

It shall be noted that the method for cryptographically transmitting information based on the calculation, in succession, of a scalar multiplication and a pairing function requires onerous processing in terms of computing for the modules being involved in the communication. Thereby, the encryption operation is made slower.

BRIEF SUMMARY OF THE INVENTION

One embodiment of the present invention is a method for processing information which is improved compared to the known methods.

One embodiment of the invention is a method of processing information to be confidentially transmitted from a first module to a second module. The method includes carrying out a first scalar multiplication in order to obtain a first result; carrying out a second scalar multiplication in order to obtain a second result; and processing the information based on the first and second results in order to obtain an encrypted information. The first scalar multiplication includes: generating a plurality of ordered factors; and calculating a plurality of first partial sums each based on a corresponding one of the ordered factors, respectively. The second scalar multiplication includes calculating a plurality of second partial sums each based on a corresponding one of the ordered factors, respectively.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

The characteristics and the advantages of the present invention will be understood from the following detailed description of one exemplary and non-limiting embodiment thereof with regard to the annexed drawings, in which:

FIG. 1 schematically shows a flow diagram referring to a method for processing information according to one embodiment of the invention;

FIGS. 2A-C schematically show flow diagrams referring to steps of the processing method from FIG. 1;

FIG. 3 schematically shows a flow diagram referring to a method for processing information according to another embodiment of the invention;

FIG. 4 schematically shows a flow diagram referring to steps of the further embodiment of the processing method from FIG. 3.

DETAILED DESCRIPTION OF THE INVENTION

It shall be noted that a method for processing information in accordance with one embodiment of the invention can be applied both to encryption protocols of this information based on the IBE identity (Identity Based Encryption) and to digital signature schemes.

Advantageously, this processing method can be applied, for example, to the IBE encryption scheme set by Boneh-Franklin in the document Identity Based Encryption from the Weil Pairing, SIAM J. Of Computing, vol. 32, No. 3, pp. 586-615, 2003, Cripto 2001, LNCS vol. 2193, Springer-Verlag 2001, pp. 213-229. Particularly, this method can be applied in the case where the encrypted information is transmitted to different scheme destinations each of which refers to a single trusted authority.

Furthermore, the processing method can be applied, for example, to the following encryption protocols: the encryption protocol set by John Malone-Lee in the document Identity-Based Signcryption, Report 2002/098, Cryptology ePrint archive, or to the one set by Divya Nalla e K. C. Reddy in the document Signcryption scheme for Identity-based Cryptosystem, Report 2003/044, Cryptology ePrint archive, or to the IBE authentication protocol set by F. Zhang, S. Liu, K. Kim in the document ID-Based One Round Authenticated Tripartite Key Agreement Protocol with Pairings, Report 2002/122, Cryptology ePrint archive.

The method provides that a processing of information may be carried out by a first module C following which a piece of information INFO can be cryptographically transmitted to a second module D.

It shall be noted that these first C and second D modules can be both hardware-type modules, i.e., electronic devices (for example, a mobile telephone, a smart card, a computer etc.), and software modules, i.e., sequences of instructions for a program.

Moreover, the method is based on mathematical schemes known to those skilled in the art and typical of the IBE protocol (Identity Based Encryption). These schemes provide:

-   -   a finite field or base field GF(q), i.e., a finite set of         elements in which addition and multiplication operations are         defined, q being an integer;     -   an extension field GF(q^(k)), i.e., a finite field containing         the base field GF(q);     -   an elliptical curve E defined on the base field GF(q).

Moreover, an odd positive integer m, co-prime with q, is defined so that the elliptical curve E contains an order point m. A number k is the smallest integer, thereby the ratio m/(q^(k)−1) is an integer.

In addition, a first m-torsion subgroup E(GF(q))[m] of the points of an elliptical curve E defined on the base field GF(q) is defined.

Particularly, said positive integer m indicates the order of the subgroup E(GF(q))[m].

Similarly, a second m-torsion subgroup of the points of an elliptical curve E defined on the extension field GF(q^(k)) has been indicated by E(GF(q^(k)))[m].

With reference to the flow diagram from FIG. 1, the method starts in step 10 from the acquisition of the information INFO (CALC1) which the first module C wants to cryptographically transmit to the second module D by means of encryption protocols based on elliptical curves.

To this purpose, as it is known to those skilled in the art, before the information INFO is encrypted and sent, the first module C calculates a first scalar multiplication and values a bilinear function. These steps have been indicated from 1 with SCAL-MLTr/m (step 12).

In greater detail, this first scalar multiplication involves a point P of the group of points of an elliptical curve E and a further random integer r. Said multiplication is carried out by adding the point P to the latter r times and has been indicated herein below with the notation [r]P.

For example, the calculation of the first scalar multiplication [r]P can be carried out from a binary representation of the random integer r, i.e.: r=Σ _(j=0) ^(t-1) r _(j)2^(j)  (1) and by expressing the result of the first scalar multiplication [r]P by: [r]P=[Σ _(j=0) ^(t-1) r _(j)2^(j) ]P=2( . . . 2(2[r _(t-1) ]P)+[r _(t-1) ]P)+ . . . )+[r ₀ ]P  (2)

It shall be noted that the operation indicated in the expression (2) can be implemented by means of an algorithm known as the Double-and Add.

Moreover, the bilinear function that is evaluated by the first module C is defined either as a map or a pairing. This pairing has been, traditionally, indicated with the notation <.,.>. For example, a known bilinear function is the Tate's pairing which, being applied to a first point P1εE(GF(q))[m] and a second point QεE(GF(q^(k)))[m], may be expressed by: <P1,Q>=f _(P1)(Q)^([(qˆk-1)/m])  (3) wherein f_(P1) indicates a rational function having coefficients in the extension field GF(q^(k)). Furthermore, the bilinear function <.,.> is applied to linearly independent m-torsion points so that this function does not take a trivial value, i.e., the unit.

Preferably, the processing method provides that the point P of the first scalar multiplication [r]P and the first point P1 of the Tate's pairing coincide with each other, i.e., P=P1εE(GF(q))[m]. This equivalence is adopted herein below for the whole description.

Furthermore, the Tate's pairing function <P1,Q>=<P,Q> can be calculated by means of a Miller's algorithm known to those skilled in the art. Particularly, this Miller's algorithm is an iterative algorithm, substantially comprising the calculation of a second scalar multiplication [m]P similar to said first multiplication [r]P. This second multiplication [m]P is carried out between the positive integer m indicative of the order of the m-torsion subgroup E(GF(q))[m] and said point P.

In addition, Miller's algorithm also comprises the iterative valuation of intermediate functions f_(v) (with v=1, 2, 3, . . . ). These intermediate functions f_(v) contribute to determine the rational function f_(P1)=f_(P) of the Tate's pairing (3) at the end of the iteration.

A preferred embodiment of the calculation step 12 of the first scalar multiplication [r]P and the valuation of the Tate's pairing <P,Q> (step SCAL-MLTr/m from FIG. 1) can be described with reference to FIGS. 2A-C.

In order to calculate the first scalar multiplication and, simultaneously, value the Tate's pairing <P,Q>, the first module C generates the further random integer r (EVAL step 20) so that the latter is smaller than the positive integer m, i.e.: 0<r<m  (4)

In an initial step of the method, the positive integer m and the further random integer r are represented in binary notation (BIN-REPR step 22), i.e., by means of a sequence of N bits (bits 0 and 1). Thereby, first (r)₂ and second (m)₂ binary numbers corresponding to the random integer r and the positive integer m, respectively, are obtained. These binary numbers (r)₂ and (m)₂ can be expressed by: (r)₂ =r _(N-1) . . . r ₁ r ₀  (5) (m)₂ =m _(N-1) . . . m ₁ m ₀  (6) wherein r₀ and m₀ are the least significant digits (or bits) of these binary numbers (r)₂ and (m)₂, whereas r_(N-1) and m_(N-1) are the most significant digits (or bits).

The method provides that said first (r)₂ and second (m)₂ binary numbers may have the same N number of bits. Therefore, given that 0<r<m, if the number of bits corresponding to the representation of the random integer r should be smaller than the number of bits being used for representing the positive integer m, the number of bits of the first binary number (r)₂ would be increased by adding one (or more) bits 0 in the positions of the sequence which refer to the most significant bits.

At this point, the first module C carries out a scanning, i.e., a bit-by-bit reading, of the first (r)₂ and second (m)₂ binary numbers. Particularly, the scanning of the first number (r)₂ is simultaneously carried out to the one of the second (m)₂ so that the bit taking the same position in the sequences of the two binary numbers are read during each step of the scanning. Moreover, this scanning is carried out from the right to the left, i.e., from the least significant bit (r₀ and m₀) to the most significant one (r_(N-1) and m_(N-1)).

Following the scanning of the least significant bits r₀ and m₀ of the first (r)₂ and second (m)₂ binary numbers (SCAN1 step 24), the first module C generates a first ordered factor Q₀ which, particularly, corresponds to the pre-determined point P (GEN1 step 26).

In general, it shall be noted that, following the scanning of each bit of the first (r)₂ and second (m)₂ binary numbers, the first module C generates a respective ordered factor. Therefore, the number of the generated factors coincides with the overall N number of bits of (r)₂ (or (m)₂).

It shall be further noted that, in accordance with one embodiment of the invention, these ordered factors Q_(i) (with i=1, 2, . . . , N−1) are obtained by carrying out N−1 doubling operations starting from the first factor Q₀, i.e., each factor Q_(i) is obtained by doubling the previous one. Thereby, a sequence S₁ of ordered factors is obtained, which can be expressed by: S₁−> Q₀=P Q₁=2P Q₂=4P Q _(N-1)=2^((N-1)) P  (7)

After the first ordered factor Q₀ has been generated, the first module C evaluates whether the least significant bits r₀ and m₀ of the first (r)₂ and second (m)₂ binary numbers are 1 or 0 (EVALL step 28).

For example, if the least significant bit r₀/m₀ of the first/second binary number (r)₂/(m)₂ is 1, the first module C calculates a first partial sum A_(1(r)) referring to the first binary number (SUM1 r step 30) and a further first partial sum A_(1(m)) referring to the second binary number (SUM1 m step 32). Particularly, these first partial sums are calculated based on the relations: A ₁₍ r)=A ₀ +Q ₀  (8) A _(1(m)) =A ₀ +Q ₀  (8′) wherein Q₀=P, whereas A₀ is traditionally fixed to the value 0. It shall be noted that (8) and (8′) are partial sums of the Double and Add algorithm.

Furthermore, a subsequent step of the processing method provides that a first intermediate function f₁ of the Tate's (EVALf1 step 34) may be evaluated.

It shall be particularly noted that in the case where the least significant bit r₀/m₀ is equal to 0 the sum (8) and/or the sum (8′) are not carried out, therefore it will be: A ₁₍ r)=A ₀=0 A _(1(m)) =A ₀=0

By proceeding with the scanning of the first/second binary number (r)₂/(m)₂, the module C calculates the second ordered factor Q₁=2P of the sequence S₁ by doubling the first factor Q₀=P (GEN2 step 38) at the bit r₁/m₁ (SCAN2 step 36).

At this point, if this bit r₁/m₁ is equal to 1 (EVAL2 step 40), a second partial sum is calculated for the first and second binary numbers (SUM2 r and SUM2 m steps 42, 44): A ₂₍ r)=A ₁₍ r)+Q ₁  (9) A _(2(m)) =A _(1(m)) +Q ₁  (9′) wherein A_(1(r)) and A_(1(m)) correspond to the partial sums being calculated in the previous step. Even in this case, if the bit r₁/m₁ is equal to 0, the sums (9) and (9′) are not carried out and the value of the second partial sum A_(2(r/m)) corresponds to the value of the sum in the previous step, i.e., A_(1(r/m)).

Moreover, the second intermediate function f₂ (EVALf2 step 46) is valued.

The procedure is similarly carried out with all the bits of the first/second binary number (r)₂/(m)₂ up to the most significant bit r_(N-1)/m_(N-1). Following the scanning of this bit (SCANn step 48) the first module C calculates the last intermediate datum Q_(N-1)=2^((N-1))P (GENn step 50). In that case, if the bit r_(N-1)/m_(N-1) is equal to 1 (EVALn step 52) the N-th partial sum A_(N(r/m)) is calculated based on the sums (SUMnr and SUMnm steps 54, 56: A _(N(r)) =A _(N-1() r)+Q _(N-1)  (10) A _(N(m)) =A _(N-1(m)) +Q _(N-1)  (10′) wherein A_(N-1(r)) and A_(N-1(m)) are the partial sums calculated in the (N−1)-th step. In the case where the most significant bits r_(N-1) and m_(N-1) are equal to 0, the N-th partial sums A_(N(r)) and A_(N(m)) take the value of the sums in the previous step, i.e., A_(N-1(r)) and A_(N-1(m)), respectively.

The N-th intermediate function f_(n) (EVALfn step 58) is also valued.

In other words, following the bit-by-bit scanning of the first (r)₂ and second (m)₂ binary numbers the first module C calculates a first and second multiplicities of partial sums. Particularly, each partial sum is calculated at one bit 1 in the binary sequences of the numbers (r)₂ and (m)₂.

It shall be further noted that the N-th partial sum calculated for the first (r)₂ and second (m)₂ binary numbers corresponds to the result of the first [r]P and second [m]P scalar multiplications, respectively.

With reference to FIG. 1, after the scalar multiplications have been carried out in accordance with the step SCAL-MLTr/m, the processing method of the invention provides the encryption (CALC2 step 14) by the first module C of the information INFO to be transmitted by means of the calculated Tate's pairing function <P,Q>.

Finally, both the encrypted information and the result of the first scalar multiplication [r]P are sent by the first module C to the second module D (INV-INF step 16).

Advantageously, the partial sums referring to the first [r]P and second [m]P scalar multiplications can be calculated starting from the same ordered factors Q_(i). In other words, the method allows the calculation of the first scalar multiplication [r]P and the one of the second multiplication [m]P to be simultaneously carried out. This considerably reduces the times of the processing being carried out by the first module C compared to the independent carrying out of each multiplication.

The cost in terms of computing to be given to the first module C both to calculate the first scalar multiplication [r]P and to value the Tate's pairing <P,Q> (by means of the Miller's algorithm) for said method can be expressed by: COST_(<P,Q>+[r]P)=└ log₂ m┘+hw(m)+hw(r)+COSTf _(V)  (11) wherein:

-   -   └log₂ m┘ is the lowest integer contained in the log₂ m. This         lowest integer corresponds to the number of doubling operations         required for building the sequence S₁ of ordered factors from         the first factor Q₀;     -   hw(m) represents the number of partial sums calculated at the         scanning of the second binary number (m)₂, hw(m) being the         hamming weight of the positive integer m, i.e., the number of 1         bits present in the second binary number (m)₂;     -   hw(r) indicates the number of partial sums calculated at the         scanning of the first binary number (r)₂, hw(r) being the number         of 1 bits present in the first binary number (r)₂;     -   COSTf_(v) is the cost in terms of computing required for the         iterative valuation of the intermediate functions f_(v).

For example, if it is assumed that the odd positive integer m=11 and the further random integer r=7, first of all, one embodiment of the invention provides that these decimal numbers may be converted into a binary base, i.e.: (7)₁₀=1*2²+1*2¹+1*2⁰−>(7)₂=111 (11)₁₀=1*2³+1*2¹+1*2⁰−>(11)₂=1011

Given that the first binary number (7)₂ includes 3 bits, whereas the second number (11)₂ includes 4, the number of bits of the first number (7)₂ is increased by adding a digit 0 in the position of the most significant bit. Thereby, the binary numbers on which the method is carried out are: (7)₂=0111 (11)₂=1011

Then, the first module C carries out the scanning of the first (7)₂ and second (11)₂ binary numbers from the least significant bit to the most significant one. Following the scanning of the least significant bits of these binary numbers (7)₂ and (11)₂, module C generates the first ordered factor Q₀=P.

Given that the least significant bits of both the binary numbers (7)₂ and (11)₂ are equal to 1, module C calculates the first partial sum A_(1(7/11)) referring to both the numbers, i.e.: A ₁₍₇₎ =A ₁₍₁₁₎=0+P=P

By proceeding with the scanning from the right to the left, the second bit of the binary numbers (7)₂ and (11)₂ is also 1. Therefore, after the second ordered factor Q₁, has been created, the first module C calculates for both the binary numbers the second partial sum A_(2(7/11)): A ₂₍₇₎ =A ₂₍₁₁₎ =P+2P=3P

The subsequent bits of the binary sequences of the first (7)₂ and second (11)₂ binary numbers are 1 and 0, respectively. Therefore, after the third intermediate datum Q₂=4P has been valued, the third partial sum A₃₍₇₎ referring to the first number (7)₂ is calculated: A ₃₍₇₎=3P+4P=7P, whereas the third partial sum A₃₍₁₁₎ referring to the second number (11)₂ is not calculated and is: A ₃₍₁₁₎ =A ₂₍₁₁₎=3P. Finally, the most significant bits of the first (7)₂ and second (11)₂ binary numbers are 0 and 1, respectively. Therefore, starting from the calculation from the last ordered factor Q₃=8P, the fourth partial sum A₄₍₇₎ referring to the first number (7)₂ is not valued by taking the value of the previous partial sum, i.e.: A ₄₍₇₎ =A ₃₍₇₎=7P, whereas, the fourth partial sum referring to the second binary number (11)₂ is calculated by: A ₄₍₁₁₎=3P+8P=11P.

It shall be further noted that, in accordance with the Miller's algorithm, module C values at the partial sums A₁₍₁₁₎, A₂ ₍₁₁₎ and A₄ ₍₁₁₎ their respective intermediate functions f₁, f₂, and f₄. On the other hand, the intermediate function f₃ corresponding to the third partial sum A₃₍₁₁₎ is not calculated.

Advantageously, with regard to the described example the first module C calculates the first scalar multiplication [7]P and values the Tate's pairing <P,Q> (including the second multiplication [11]P) by carrying out a number of operations which is smaller than the ones required if the known processing methods should be employed. In fact, from the expression (11), for r=7 and m=11 the number of the overall operations to be carried out with the method, i.e., the cost in terms of computing of this method is: $\begin{matrix} {{COST}_{{< P},{Q > {{+ {\lbrack r\rbrack}}P}}} = {{3\quad({doubling})} + {3\quad({sums})} + {3\quad({sums})} + {{COST}\quad f_{v}}}} \\ {= {{3\quad({doubling})} + {6\quad({sums})} + {{COST}\quad{f_{v}.}}}} \end{matrix}$

Therefore, the operations for the first module C are: three doubling operations, six partial sums and the valuation of the intermediate functions f_(v).

It shall be particularly noted that with the preferred embodiment of the method for processing information the number of required doubling operations is smaller than the one of the known methods. In fact, these latest known methods provide that two independence sequences of ordered factors may be calculated, one of which is used for calculating the first scalar multiplication [r]P, whereas the other one is used for calculating the second multiplication [m]P.

For example, for r=7 and m=11, with the known processing methods, the number of required doubling operations would be equal to five.

As it is known to those skilled in the art, in order to calculate the first scalar multiplication [r]P with the Double-and-Add algorithm, the scanning of the first binary number (r)₂ may be also carried out from the most significant bit r_(N-1) to the least significant one r₀, i.e., from the left to the right of the binary sequence. However, in this case, the calculation of the first multiplication [r]P would be onerous from the computing viewpoint because the first module C should carry out a larger number of operations compared to the ones required for the scanning of the same number (r)₂ from the right to the left.

On the contrary, the calculation of the second scalar multiplication [m]P being included in the Miller's algorithm would require a smaller number of calculations in the case where the scanning of the second binary number (m)₂ is carried out from the most significant bit m_(N-1) to the least significant one m₀, i.e., from the left to the right. In this case, the carrying out of the Miller's algorithm would be faster.

Having said that, a further embodiment of the processing method provides that the first module C may carry out the scanning of the second binary number (m)₂ from the most significant bit m_(N-1) to the least significant one m₀ (from the left to the right).

Moreover, in order to simultaneously carry out the scanning of the first binary number (r)₂ from the right to the left by making use of the advantage which this entails in terms of computing, the first module C provides for advantageously representing said first number (r)₂ in a new numerical base or dynamic base B_(m).

Particularly, this dynamic base B_(m) is generated from the binary representation of the odd positive integer m, i.e., from (m)₂. In fact, the positive integer m can be expressed by: m=2^(N-1)+2^(d) ^(n-2) +2^(d) ^(n-2) + . . . +2^(d) ¹ +1  (12) wherein the exponent d_(i) indicates the position of the bits 1 within the binary sequence of m and the index h represents the hamming weight of the integer m.

From the expression (12), the dynamic base B_(m) being function of the integer m can be expressed by: $\begin{matrix} \begin{matrix} {B_{m} = \left\{ 2^{i} \right.} & {{{with}\quad 0} \leq i \leq {N - 1 - d_{h - 2}}} \\ {2^{i}\left( {2^{{({N - 1})} - {(d_{h - 2})}} + 1} \right)} & {{{with}\quad 0} \leq i \leq {d_{h - 2} - d_{h - 3}}} \\ {2^{i}\left( {\begin{matrix} 2^{{(d_{h - 2})} - {(d_{h - 3})}} \\ \left( {2^{{({N - 1})} - {(\quad d_{h - 2})}} + 1} \right) \end{matrix} + 1} \right)} & {{{with}\quad 0} \leq i \leq {d_{h - 3} - d_{h - 4}}} \\ \ldots & \quad \\ {\left. {{2^{d_{1}}\left( {{\ldots\begin{pmatrix} \begin{matrix} 2^{d_{h - 3} - d_{h - 2}} \\ \left( {2^{N - 1 - d_{h - 2}} + 1} \right) \end{matrix} \\ {+ 1} \end{pmatrix}}\ldots} \right)} + 1} \right\}.} & \quad \end{matrix} & (13) \end{matrix}$

It shall be noted that the dimension w of the dynamic base B_(m), i.e., the number of elements of this base is given by: w=└ log₂(m)┘+hw(m).  (14)

Furthermore, given the positive integer m, the highest number which can be represented by means of the dynamic base B_(m) being built from m is indicated by max_record. In this case, if the random integer r≦max_record, this number r can be represented as a linear combination of the elements of the dynamic base B_(m), i.e.: r=c _(w-1) b _(w-1) + . . . +c ₁ b ₁ +c ₀ b ₀  (15) wherein c_(i)ε{0,1} and b_(i)εB_(m).

In accordance with the equivalence (15), the random integer r may be re-coded by means of the dynamic base B_(m) and this re-coding is indicated by re-cod(r) in order to be distinguished from the binary representation of the random integer r.

It shall be further noted that the relation (4) is also valid with reference to the second embodiment of the method. This means that in (15) is c_(w-1)=0.

For example, for r=7 and m=11, the further embodiment of the processing method can be described with reference to FIG. 3.

Particularly, this embodiment of the method provides, first of all, that the dynamic base B_(m)=B₁₁ (GEN B_(m) step 70) may be generated from the first module C. This base B₁₁ is generated in accordance with (13) and from the scanning of the binary number (11)₂, i.e. (11)₂ =m ₃ m ₂ m ₁ m ₀=1011 from the most significant bit m₃ to the least significant one m₀.

It shall be noted that in accordance with (13), the first element of the dynamic base B₁₁ (if this base is read from the right to the left) is always equal to 1. This first element is generated by the first module C following the scanning of the most significant bit m₃=1 of the binary number (11)₂.

Following the scanning of the bit m₂=0 of (11)₂, the first module C doubles the first element of the base B₁₁, so that the second element is equal to 2.

With the scanning of the subsequent bit m₁=1 of (11)₂, the first module C doubles the value of the previous element of the base (i.e., of the second equal to 2), thereby the third element of the base B₁₁ is equal to 4. Moreover, as the read bit m₁=1, the first module C generates the fourth element of the base B₁₁ by increasing this third element by one unit. After all, the fourth element of the base B₁₁ is equal to 5.

Similarly, following the scanning of the bit m₀=1 of (11)₂, the first module C doubles the previous element, i.e., the fourth being equal to 5. Therefore, the fifth element of the base B₁₁ is equal to 10.

Furthermore, given that the subsequent bit being read during the scanning of the number (11)₂ is m₀=1, the first module C generates the sixth element of the base B₁₁ by increasing the value of this fifth element by one unit. Thereby, the sixth element of the base B₁₁ is fixed to 11.

After all, the dynamic base generated by the first module C is: B₁₁={11,10,5,4,2,1}.

It shall be noted that the random integer r=7 can be thus represented in this base B₁₁ by using known re-coding algorithms (RECODr step 72). For example, the following re-coding algorithm can be used: $\begin{matrix} {t = {{r - 11} = {{7 - 11} \geq {{0?{No}}\quad\text{->}\quad{then}}}}} & {{{re}\text{-}{{cod}(r)}} = \left\lbrack {0\quad{xxxxx}} \right\rbrack} \\ {t = {{r - 10} = {{7 - 10} \geq {{0?{No}}\quad\text{->}\quad{then}}}}} & {{{re}\text{-}{{cod}(r)}} = \left\lbrack {0\quad 0{xxxx}} \right\rbrack} \\ {{t = {{r - 5} = {{7 - 5} = {{2 \geq {{0?{Yes}}\quad\text{->}\quad{then}\quad r}} = t}}}};} & {{{re}\text{-}{{cod}(r)}} = \left\lbrack {0\quad 01{xxx}} \right\rbrack} \\ {t = {{r - 4} = {{2 - 4} \geq {{0?{No}}\quad\text{->}\quad{then}}}}} & {{{re}\text{-}{{cod}(r)}} = \left\lbrack {0\quad 010{xx}} \right\rbrack} \\ {{t = {{r - 2} = {{2 - 2} = {{0 \geq {{0?{Yes}}\quad\text{->}\quad{then}\quad r}} = t}}}};} & {{{re}\text{-}{{cod}(r)}} = \left\lbrack {0\quad 0101x} \right\rbrack} \\ {t = {{r - 1} = {{0 - 1} \geq {{0?{No}}\quad\text{->}\quad{then}}}}} & {{{re}\text{-}{{cod}(r)}} = \left\lbrack {0\quad 01010} \right\rbrack} \end{matrix}$ and the random number r=7 is re-coded in the base B₁₁ by: re-cod(r)=re-cod(7)=001010 by particularly indicating each bit of this re-coded number: re-cod(7)=r ₅ r ₄ r ₃ r ₂ r ₁ r ₀=001010.

At this point, the first module C may carry out the first [r]P and second [m]P scalar multiplications (SCAL-MLT1 r/m step 74) by simultaneously scanning the binary number (11)₂ from the left to the right and the re-coded number re-cod(7) from the right to the left.

With reference to FIG. 4, a carrying out example of the SCAL-MLT1 r/m step 74 has been described with a flow diagram.

Particularly, the first bit of the binary number (11)₂ (the bit m₃ in the examined example) is always equal to 1, at the first step of the scanning (PH0 step 80). The first module C values the first ordered factor Q₀=P which coincides with the first partial sum: A ₁₍₁₁₎ =A ₀ +Q ₀ =P (A₀=0 traditionally) carried out for this binary number (11)₂.

On the other hand, the first bit of the number re-cod(7) is r₀=0, therefore, the corresponding first partial sum is not valued, i.e., A₁₍₇₎=0.

The following step of the scanning (PH1 step 82), provides that the second ordered factor Q₁=2Q₀=2P being obtained by doubling this first factor Q₀ may be generated. In this case, with r₁=1, the second partial sum referring to the number re-cod(7) is carried out from the partial sum being valued in the previous step (PH0 step 80), i.e.: A ₂₍₇₎ =A ₁₍₇₎ +Q ₁=2P On the other hand, with m₂=0 the second partial sum referring to the number (11)₂ is not carried out.

In the subsequent step of the scanning (PH2 step 84), the first module C values the third ordered factor Q₂=2Q₁=4P. Given that r₂=0 the third partial sum for re-cod(7) (which is equal to the previous one, i.e., A₃₍₇₎=A₂₍₇₎) is not valued.

On the other hand, with m₁=1 the fourth ordered factor Q₃=Q₂+P=5P is generated by increasing the third datum Q₂ by one unit.

The subsequent bit of the number re-cod(7) is r₃=1. From this factor Q₃, the partial sum is obtained: A ₄₍₇₎ =A ₃₍₇₎ +Q ₃=7P.

In the subsequent step of the scanning (PH3 step 86), the fifth ordered factor Q₄=2Q₃=10P is valued. Given that the bit r₄=0 the partial sum for the number re-cod(7) which is equal to the one being valued in the previous step is not calculated.

On the other hand, with m₀=1, the sixth ordered factor Q₅=Q₄+P=11P is valued. This factor Q₅ coincides with the last partial sum for the binary number (11)₂ and, particularly, with the final result of the second scalar multiplication [m]P. On the other hand, the last partial sum calculated for re-cod(7), i.e., A₄₍₇₎=7P coincides with the result of the first scalar multiplication [r]P.

It shall be noted that the cost in terms of computing for calculating the first [r]P and second [m]P multiplications can be expressed by: COST_([m]P+[r]p)=└ log₂ m┘+hw(m)+hw(re-cod(r))  (16) wherein:

-   -   −└log₂ m┘ is the number of doubling operations required for         calculating the ordered factors Q_(i) from the first factor Q₀;     -   hw(m) is the number of partial sums calculated at the scanning         of the second binary number (m)₂, i.e., the number of bits 1         being present in the second binary number (m)₂;     -   hw(re-cod(r)) indicates the number of partial sums calculated at         the scanning of the re-coded number re-cod(r), i.e., the number         of bits 1 being present in this number.

In accordance with the further embodiment of the method, the first module C may calculate the first scalar multiplication [r]P (through the scanning of the re-coded binary number re-cod(r) from the right to the left) and value the Tate's pairing <P,Q> by means of the Miller's algorithm (scanning of the binary number (m)₂ from the left to the right). In this case, the cost in terms of computing of this operation is: COST_(<P,Q>+[r]p)=└ log₂ m┘+hw(m)+hw(re-cod(r))+Costf _(v)  (17)

Finally, the first module C may also simultaneously value a multiplicity of Tate's pairing functions. For example, in the case of two pairing functions <P,R> and <P,Q>, the cost in terms of computing in order to calculate these functions by means of the Miller algorithm can be expressed by: COST_(<P,Q>+<P,R)>=└ log₂ m┘+hw(m)+Costf _(v)+Costf _(W)  (18) It has been noted that the calculation of the pairing functions <P,R> and <P,Q> in accordance with the processing method described above, i.e., by using the same doubling operations, reduces the cost in terms of computing by about 50%.

Finally, it shall be noted that with reference to FIG. 3, after the scalar multiplications have been carried out in accordance with the step SCAL-MLT1 r/m, the processing method of the invention provides the encryption (CALC step 76) from the first module C of the information INFO to be transmitted by means of the calculated Tate's pairing function <P,Q>.

This encrypted information and the result of the first scalar multiplication [r]P are then sent from the first module C to the second D (INV-INF1 step 78).

Appendix

The listings of the programs referring to the known Double and Add algorithms (algorithm 1: scanning right->left and algorithm 2: scanning left->right) and the Miller's algorithms (algorithm 3: scanning right->left and algorithm 4: scanning left->right) are set forth herein below by way of example. Algorithm 1 Input: t = ┌log2 r┐ r = (rt−1 , ..., r0)2, rt−1 = 1 P Output: [r]P V := P; for i = t−2 down to 0 do V := [2]V; if ri = 1 then V := V + P; return V

Algorithm 2 Input: t = ┌log2 r┐ r = (rt−1 , ..., r0)2, P Output: [r]P V := P; if (r0 = 1 ) then Z := P else Z := OE; for i = 1 to t−1 do V := [2]V; if (ri = 1 ) then Z := Z + V; return Z

Algorithm 3 Input: t = ┌log2 m┐ m = (mt−1, ..., m0)2, mt−1 = m0 = 1 P, Q, q, k Output: <P,Q> = fP(DQ)[(q{circumflex over ( )}k −1)/m] fV := 1; V := P; for i = t−2 down to 1 do Vnew := [2]V; fV := fV2 · gV,V(DQ) / g2V(DQ): V := Vnew; if (mi = 1) then Vnew := V+P; fV := fV · gV,P(DQ) / gV+P(DQ); V := Vnew; fV := fV2 · gV,V(DQ) / g2V(DQ); fV := fV[(q{circumflex over ( )}k −1)/m]; return fV

Algorithm 4 Input: t = ┌log2 m┐ m = (mt−1, ..., m0)2, mt−1 = m0 = 1 P, Q, q, k Output: <P,Q> = fP(DQ)[(q{circumflex over ( )}k −1)/m] V = W := P; fV = fW := 1; for i = 1 to t−2 do Vnew := [2]V; fV := fV2 · gV,V(DQ) / g2V(DQ) V := Vnew; if (mi = 1) then Wnew := W+V; fW := fV · fW · gV,W(DQ) / gV+W(DQ) W := Wnew; Vnew := [2]V; fV := fV2 · gV,V(DQ) / g2V(DQ) V := Vnew; fW := fV fW; fW := fW·[(q{circumflex over ( )}k −1)/m]; return fW;

Furthermore, the listings of examples for programs referring to the method for processing information described above (algorithm 5: scanning right->left for r and m), and the further embodiment of the method (algorithm 6 and algorithm 7: scanning right->left for re-cod(r) and scanning left->right for m) are shown herein below. Finally, the algorithm 8 relates to the simultaneous calculation of two pairing functions. Algorithm 5 Input: t = ┌log2 m┐ m = (mt−1, ..., m0)2, mt−1 = m0 = 1 s = ┌log2 r┐ r = (rs−1, ..., r0)2, rs−1 = r0 = 1 P, Q, q, k Output: [r]P, <P,Q> = fP(DQ)[(q{circumflex over ( )}k −1)/m] V = W = Z := P; fV = fW := 1; if (t−s > 0) then for i = s to t−1 ri = 0; for i = 1 to t−2 do Vnew = [2]V; fV := fV2 · gV,V(DQ) / g2V(DQ); V := Vnew; if (mi = 1) then Wnew := W+V; fW := fV · fW · gV,W(DQ) / gV+W(DQ); // fV · fW is a mult. in GF(qk) W := Wnew; if (ri = 1) then Z := Z+V; fV := fV2 · gV,V(DQ) / g2V(DQ); V := [2]V; fW := fV · fW; fW := fW·[(q{circumflex over ( )}k −1)/m]; if (rt−1 = 1) Z := Z+V; for i = t to s−1 do V := [2]V; if (ri = 1) then Z := Z+V; return Z, fW;

Algorithm 6 Input: t = ┌log2 m┐ m = (mt−1, ..., m0)2, mt−1 = m0= 1, w:= log2(m) + hw(m) −1 re-cod(r) = (cw−1, ...., c0)2 P Output: [r]P, [m]P V = P; if (c0 = 1) then Z = P; else Z = OE; j = 1; // index of the scanner bit of re-cod(r) for i = t − 2 down to 0 do //index of the bit of m V = [2]V; if (cj = 1) then Z = Z + V; if (mi = 1) then j⁺⁺ V = V + P; if (cj−1 = 1) then Z = Z + V; j⁺⁺ if (cj = 1) then Z = Z + V; return Z, V;

Algorithm 7 Input: t = ┌log2 m┐ m = (mt−1, ..., m0)2, mt−1 = m0 = 1 re-cod(r) = (cw−1, .... c0)2, P, Q, q, k, Output: [r]P, <P,Q> = fP(DQ)[(q{circumflex over ( )}k −1)/m] fV := 1; V := P; if (c0= 1) then Z = P; else Z = OE; j = 1; // j is the index of the re-cod(r) bit for i = t−2 down to 1 do begin Vnew := [2]V fV := fV2 · gV,V(DQ) / g2V(DQ) ; V := Vnew; if (cj = 1) then Z = Z + V; if (mi = 1) then j⁺⁺ Vnew := V + P; fV := fV · gV,P(DQ) / gV+P(DQ); V := Vnew; if (cj = 1) then Z = Z + V; j⁺⁺ if (cj = 1) then Z = Z + V; fV := fV2 · gV,V(DQ) / g2V(DQ) ; fV := fV[(q{circumflex over ( )}k −1)/m]; return Z, fV

Algorithm 8 Input: t = ┌log2 m┐ m = (mt−1, ..., m0)2, mt−1 = m0 = 1 P, Q, R, q, k, Output: <P,Q> = fP(DQ)[(q{circumflex over ( )}k −1)/m], <P,R> = fP(R)[(q{circumflex over ( )}k −1)/m] fV = fw := 1; V := P; for i = t−2 down to 1 do Vnew := [2]V fV := fV2 · gV,V(DQ) / g2V(DQ) ; fW := fW2 · gV,V(DR) / g2V(DR) ; V := Vnew; if (mi = 1) then Vnew := V + P; fV := fV · gV,P(DQ) / gV+P(DQ); fw := fw · gV,P(DR) / gV+P(DR); V := Vnew; fV := fV2 · gV,V(DQ) / g2V(DQ) ; fW := fW2 · gV,V(DR) / g2V(DR) ; fV := fV[(q{circumflex over ( )}k −1)/m]; fW := fW[(q{circumflex over ( )}k −1)/m]; return fV, fW

Obviously, aiming at satisfying contingent and specific needs, those skilled in the art will be able to carry out further modifications and variants, all of them being contemplated within the scope of protection of the invention, such as defined in the claims below.

All of the above U.S. patents, U.S. patent application publications, U.S. patent applications, foreign patents, foreign patent applications and non-patent publications referred to in this specification and/or listed in the Application Data Sheet, are incorporated herein by reference, in their entirety. 

1. A method of processing information to be confidentially transmitted from a first module to a second module, said method comprising the steps of: carrying out a first scalar multiplication in order to obtain a first result, said first scalar multiplication including: generating a plurality of ordered factors; and calculating a plurality of first partial sums each based on a corresponding one of the ordered factors, respectively; carrying out a second scalar multiplication in order to obtain a second result, said second scalar multiplication including calculating a plurality of second partial sums each based on a corresponding one of the ordered factors, respectively; and processing said information based on the first and second results in order to obtain an encrypted information.
 2. The method in accordance with claim 1, wherein the second scalar multiplication is carried out between a point of an elliptical curve and a positive integer indicative of an order of a point of the curve and wherein the first scalar multiplication is carried out between the point and a random integer generated by the first module.
 3. The method in accordance with claim 2, wherein generating the plurality of ordered factors includes generating a first ordered factor, and generating a subsequent, second ordered factor by doubling the first ordered factor, the first ordered factor being coincident with the point of the elliptical curve.
 4. The method in accordance with claim 3, wherein the first and second scalar multiplications comprise the steps of: representing, from the first module, said random integer and said positive integer by sequences of bits in order to obtain a first and second binary numbers, respectively; carrying out a simultaneous reading of a first bit of said first binary number and a first bit of said second binary number, said first bits being a least significant bit in their respective binary numbers; verifying whether the first bits of the first and second binary numbers are a first logic value; building the first and second partial sums in response to verifying that the first bits of the first and second binary numbers are the first logic value; repeating the carrying out, verifying, and building steps for each bit of the first and second binary numbers.
 5. The method in accordance with claim 4, wherein the first and second partial sums using the least significant bits of the first and second binary numbers coincide with said first ordered factor.
 6. The method in accordance with claim 5, wherein the first and second partial sums using respective bits of the binary numbers following the least significant bits are calculated by adding the ordered factor generated for the respective bits of the binary numbers following the least significant bits to the partial sums for the least significant bits.
 7. The method in accordance with claim 4, wherein said building step comprises the further step of evaluating a first intermediate function referring to a pairing function.
 8. The method in accordance with claim 7, wherein said pairing function is Tate's pairing.
 9. The method in accordance with claim 4, having a cost in terms of computing equal to: COST_(<P,Q≦+[r]P)=└ log₂ m┘+hw(m)+hw(r)+COSTf _(v) with: └log₂ m┘ being a smallest integer included in the log₂ m; hw(m) being the number of partial sums calculated from the reading of the second binary number, wherein hw(m) is the number of bits of the second binary number having the first logic value; hw(r) being the number of partial sums calculated from the reading of the first binary number wherein hw(r) is the number of bits of the first binary number having the first logic value; COSTf_(v) being the cost in terms of computing for the verifying steps.
 10. The method in accordance with claim 4, comprising the step of generating a dynamic numerical base from said second binary number, said dynamic base being valued by an expression: $\begin{matrix} {B_{m} = \left\{ 2^{i} \right.} & {{{with}\quad 0} \leq i \leq {N - 1 - d_{h - 2}}} \\ {2^{i}\left( {2^{{({N - 1})} - {(d_{h - 2})}} + 1} \right)} & {{{with}\quad 0} \leq i \leq {d_{h - 2} - d_{h - 3}}} \\ {2^{i}\left( {\begin{matrix} 2^{{(d_{h - 2})} - {(d_{h - 3})}} \\ \left( {2^{{({N - 1})} - {(\quad d_{h - 2})}} + 1} \right) \end{matrix} + 1} \right)} & {{{with}\quad 0} \leq i \leq {d_{h - 3} - d_{h - 4}}} \\ \ldots & \quad \\ \left. {{2^{d_{1}}\left( {{\ldots\begin{pmatrix} \begin{matrix} 2^{d_{h - 3} - d_{h - 2}} \\ \left( {2^{N - 1 - d_{h - 2}} + 1} \right) \end{matrix} \\ {+ 1} \end{pmatrix}}\ldots} \right)} + 1} \right\} & \quad \end{matrix}$ said dynamic base allowing a re-coding of the first binary number to be carried out in order to obtain a re-coded number.
 11. The method in accordance with claim 10, further comprising carrying out further first and further second scalar multiplications by: carrying out a simultaneous reading of one most significant bit of said second binary number and one least significant bit of said re-coded number; generating a third ordered factor for said most significant bit read, said third ordered factor being equal to the point of the elliptical curve; building the first partial sum for said most significant bit and the second partial sum for said least significant bit of said re-coded number following a verification that said most significant bit and said least significant bit of said re-coded number have the first logic value; repeating the carrying out, generating, and building steps for each bit of the second binary number and the re-coded number, wherein repeating the generating step includes generating ordered factors subsequent to said third ordered factor either by adding the point to a previous one of the ordered factors or by doubling said previous ordered factor.
 12. The method in accordance with claim 1, wherein the first and second modules are both electronic devices.
 13. A system for processing information, the system comprising: first scalar multiplication means for carrying out a first scalar multiplication in order to obtain a first result, said first scalar multiplication including: generating means for generating a plurality of ordered factors; and means for calculating a plurality of first partial sums each based on a corresponding one of the ordered factors, respectively; second scalar multiplication means for carrying out a second scalar multiplication in order to obtain a second result, said second scalar multiplication means including means for calculating a plurality of second partial sums each based on a corresponding one of the ordered factors, respectively; and means for processing said information based on the first and second results in order to obtain an encrypted information.
 14. The system of claim 13, wherein the second scalar multiplication means carries out the second scalar multiplication between a point of an elliptical curve and a positive integer indicative of an order of a point of the curve and wherein the first scalar multiplication is carried out between the point and a random integer generated by the first module.
 15. The system of claim 14, wherein the generating means generate a first ordered factor, and generate a subsequent, second ordered factor by doubling the first ordered factor, the first ordered factor being coincident with the point of the elliptical curve.
 16. The system of claim 14, wherein the first and second scalar multiplication means include: means for representing said random integer and said positive integer by sequences of bits in order to obtain a first and second binary numbers, respectively; means for carrying out a simultaneous reading of a first bit of said first binary number and a first bit of said second binary number, said first bits being a least significant bit in their respective binary numbers; means for verifying whether the first bits of the first and second binary numbers are a first logic value; means for building the first and second partial sums in response to verifying that the first bits of the first and second binary numbers are the first logic value; means for repeating the carrying out, verifying, and building for each bit of the first and second binary numbers.
 17. The system of claim 16, wherein the first and second partial sums using the least significant bits of the first and second binary numbers coincide with said first ordered factor.
 18. The system of claim 17, wherein the first and second partial sums using respective bits of the binary numbers following the least significant bits are calculated by adding the ordered factor generated for the respective bits of the binary numbers following the least significant bits to the partial sums for the least significant bits.
 19. The system of claim 16, comprising means for generating a dynamic numerical base from said second binary number, said dynamic base being being valued by an expression: $\begin{matrix} {B_{m} = \left\{ 2^{i} \right.} & {{{with}\quad 0} \leq i \leq {N - 1 - d_{h - 2}}} \\ {2^{i}\left( {2^{{({N - 1})} - {(d_{h - 2})}} + 1} \right)} & {{{with}\quad 0} \leq i \leq {d_{h - 2} - d_{h - 3}}} \\ {2^{i}\left( {\begin{matrix} 2^{{(d_{h - 2})} - {(d_{h - 3})}} \\ \left( {2^{{({N - 1})} - {(\quad d_{h - 2})}} + 1} \right) \end{matrix} + 1} \right)} & {{{with}\quad 0} \leq i \leq {d_{h - 3} - d_{h - 4}}} \\ \ldots & \quad \\ \left. {{2^{d_{1}}\left( {{\ldots\begin{pmatrix} \begin{matrix} 2^{d_{h - 3} - d_{h - 2}} \\ \left( {2^{N - 1 - d_{h - 2}} + 1} \right) \end{matrix} \\ {+ 1} \end{pmatrix}}\ldots} \right)} + 1} \right\} & \quad \end{matrix}$ said dynamic base allowing a re-coding of the first binary number to be carried out in order to obtain a re-coded number.
 20. The system of claim 19, further comprising means for carrying out further first and further second scalar multiplications by: carrying out a simultaneous reading of one most significant bit of said second binary number and one least significant bit of said re-coded number; generating a third ordered factor for said most significant bit read, said third ordered factor being equal to the point of the elliptical curve; building the first partial sum for said most significant bit and the second partial sum for said least significant bit of said re-coded number following a verification that said most significant bit and said least significant bit of said re-coded number have the first logic value; repeating the carrying out, generating, and building steps for each bit of the second binary number and the re-coded number, wherein repeating the generating step includes generating ordered factors subsequent to said third ordered factor either by adding the point to a previous one of the ordered factors or by doubling said previous ordered factor.
 21. A computer-readable medium, comprising code that, when loaded into a memory of a computer, causes the computer to process information to be confidentially transmitted from a first module to a second module by a method that includes: carrying out a first scalar multiplication in order to obtain a first result, said first scalar multiplication including: generating a plurality of ordered factors; and calculating a plurality of first partial sums each based on a corresponding one of the ordered factors, respectively; carrying out a second scalar multiplication in order to obtain a second result, said second scalar multiplication including calculating a plurality of second partial sums each based on a corresponding one of the ordered factors, respectively; and processing said information based on the first and second results in order to obtain an encrypted information.
 22. The computer-readable medium of claim 21, wherein the second scalar multiplication is carried out between a point of an elliptical curve and a positive integer indicative of an order of a point of the curve and wherein the first scalar multiplication is carried out between the point and a random integer generated by the first module.
 23. The computer-readable medium of claim 2, wherein generating the plurality of ordered factors includes generating a first ordered factor, and generating a subsequent, second ordered factor by doubling the first ordered factor, the first ordered factor being coincident with the point of the elliptical curve.
 24. The computer-readable medium of claim 23, wherein the first and second scalar multiplications comprise the steps of: representing, from the first module, said random integer and said positive integer by sequences of bits in order to obtain a first and second binary numbers, respectively; carrying out a simultaneous reading of a first bit of said first binary number and a first bit of said second binary number, said first bits being a least significant bit in their respective binary numbers; verifying whether the first bits of the first and second binary numbers are a first logic value; building the first and second partial sums in response to verifying that the first bits of the first and second binary numbers are the first logic value; repeating the carrying out, verifying, and building steps for each bit of the first and second binary numbers.
 25. The computer-readable medium of claim 24, wherein the first and second partial sums using the least significant bits of the first and second binary numbers coincide with said first ordered factor.
 26. The computer-readable medium of claim 25, wherein the first and second partial sums using respective bits of the binary numbers following the least significant bits are calculated by adding the ordered factor generated for the respective bits of the binary numbers following the least significant bits to the partial sums for the least significant bits.
 27. The computer-readable medium of claim 24, wherein the method includes generating a dynamic numerical base from said second binary number, said dynamic base being valued by an expression: $\begin{matrix} {B_{m} = \left\{ 2^{i} \right.} & {{{with}\quad 0} \leq i \leq {N - 1 - d_{h - 2}}} \\ {2^{i}\left( {2^{{({N - 1})} - {(d_{h - 2})}} + 1} \right)} & {{{with}\quad 0} \leq i \leq {d_{h - 2} - d_{h - 3}}} \\ {2^{i}\left( {\begin{matrix} 2^{{(d_{h - 2})} - {(d_{h - 3})}} \\ \left( {2^{{({N - 1})} - {(\quad d_{h - 2})}} + 1} \right) \end{matrix} + 1} \right)} & {{{with}\quad 0} \leq i \leq {d_{h - 3} - d_{h - 4}}} \\ \ldots & \quad \\ \left. {{2^{d_{1}}\left( {{\ldots\begin{pmatrix} \begin{matrix} 2^{d_{h - 3} - d_{h - 2}} \\ \left( {2^{N - 1 - d_{h - 2}} + 1} \right) \end{matrix} \\ {+ 1} \end{pmatrix}}\ldots} \right)} + 1} \right\} & \quad \end{matrix}$ said dynamic base allowing a re-coding of the first binary number to be carried out in order to obtain a re-coded number.
 28. The computer-readable medium of claim 27, wherein the method includes carrying out further first and further second scalar multiplications by: carrying out a simultaneous reading of one most significant bit of said second binary number and one least significant bit of said re-coded number; generating a third ordered factor for said most significant bit read, said third ordered factor being equal to the point of the elliptical curve; building the first partial sum for said most significant bit and the second partial sum for said least significant bit of said re-coded number following a verification that said most significant bit and said least significant bit of said re-coded number have the first logic value; repeating the carrying out, generating, and building steps for each bit of the second binary number and the re-coded number, wherein repeating the generating step includes generating ordered factors subsequent to said third ordered factor either by adding the point to a previous one of the ordered factors or by doubling said previous ordered factor. 